Tarzan plans to cross a gorge by swinging in an arc from a hanging vine. If his arms are capable of exerting a force of 1170 N on the rope, what is the maximum speed (in meters/second) he can tolerate at the lowest point of his swing? His mass is 83.0 kg, and the vine is 18.0 m long.
his weight plus the centripetal force must equal 1170 N
83.0 [g + (v^2 / r)] = 1170
v^2 / 18.0 = (1170 / 83.0) - 9.81
Well, Tarzan is really swinging into action! Let's crunch some numbers here.
First, we need to find the tension in the vine when Tarzan is at the lowest point of his swing. At the lowest point, Tarzan's weight is balanced out by the tension in the vine. So we have:
Weight = Tension
Using the equation for weight, which is mass multiplied by gravitational acceleration (mg), we can find the weight of Tarzan:
Weight = mass × gravitational acceleration
Weight = 83.0 kg × 9.8 m/s²
Weight = 813.4 N
Now, since Tarzan exerts a force of 1170 N on the vine, the tension in the vine will also be 1170 N.
Next, let's tackle the speed at the lowest point of the swing. At the lowest point, all of Tarzan's initial potential energy is converted into kinetic energy. We can find the initial potential energy using the equation:
Potential Energy = mass × gravitational acceleration × height
Since the height is half the length of the vine, which is 18.0 m divided by 2, we get:
Potential Energy = 83.0 kg × 9.8 m/s² × (18.0 m / 2)
Potential Energy = 7277.4 J
Since the potential energy is converted entirely into kinetic energy, we have:
Kinetic Energy = Potential Energy
½ × mass × velocity² = 7277.4 J
Rearranging the equation to solve for velocity:
Velocity = √((2 × Kinetic Energy) / mass)
Plugging in the numbers:
Velocity = √((2 × 7277.4 J) / 83.0 kg)
Velocity ≈ √175.2 m²/s²
Velocity ≈ 13.2 m/s
So Tarzan's maximum speed at the lowest point of his swing would be approximately 13.2 meters per second. That's quite the wild ride! Just make sure to hold on tight, Tarzan!
To find the maximum speed that Tarzan can tolerate at the lowest point of his swing, we need to consider the forces acting on him.
1. First, let's calculate the gravitational force acting on Tarzan:
Gravitational force (Fg) = mass × acceleration due to gravity
= 83.0 kg × 9.8 m/s^2
= 813.4 N
2. Next, we need to calculate the tension in the vine at the lowest point of the swing. At the lowest point, the net force acting on Tarzan is equal to the centripetal force.
Net force (Fnet) = centripetal force (Fc)
Centripetal force (Fc) = mass × (velocity^2 / radius)
= 83.0 kg × (velocity^2 / 18.0 m)
The tension in the vine is equal to the net force:
Tension (T) = Fnet = Fc = mass × (velocity^2 / radius)
3. Therefore, we can set up an equation:
Fg = T + Fnet
Substituting the values we have:
813.4 N = T + (83.0 kg × (velocity^2 / 18.0 m))
4. Rearranging the equation to solve for the velocity:
T + (83.0 kg × (velocity^2 / 18.0 m)) = 813.4 N
(83.0 kg × (velocity^2 / 18.0 m)) = 813.4 N - T
(velocity^2 / 18.0 m) = (813.4 N - T) / 83.0 kg
velocity^2 = 18.0 m × (813.4 N - T) / 83.0 kg
velocity = sqrt((18.0 m × (813.4 N - T)) / 83.0 kg)
5. Finally, we can substitute the tension value:
velocity = sqrt((18.0 m × (813.4 N - 1170 N)) / 83.0 kg)
Using a calculator, we can now solve for the maximum speed (velocity).
To find the maximum speed Tarzan can tolerate at the lowest point of his swing, we need to consider the conservation of mechanical energy.
The mechanical energy of a swinging object consists of its potential energy (PE) and kinetic energy (KE). At the highest point of the swing, Tarzan's kinetic energy is zero, and all of his mechanical energy is in the form of potential energy. As he swings down, his potential energy is gradually converted into kinetic energy until he reaches the lowest point of the swing, where all of his mechanical energy is in the form of kinetic energy.
We can calculate the potential energy at the highest point of Tarzan's swing using the formula:
PE = m * g * h
where m is Tarzan's mass (83.0 kg), g is the acceleration due to gravity (9.8 m/s^2), and h is the height above the lowest point (which is equal to the length of the vine, 18.0 m).
PE = 83.0 kg * 9.8 m/s^2 * 18.0 m
= 14,532 J
At the highest point, this potential energy is converted entirely into kinetic energy at the lowest point. Thus, we can equate the potential energy to the kinetic energy:
KE = 14,532 J
The kinetic energy formula is given by:
KE = 1/2 * m * v^2
where v is the speed we want to find.
Rearranging the formula, we can solve for v:
v^2 = 2 * KE / m
v^2 = 2 * 14,532 J / 83.0 kg
v^2 = 349.88 m^2/s^2
Taking the square root of both sides:
v = √349.88 m^2/s^2
v ≈ 18.70 m/s
Therefore, the maximum speed Tarzan can tolerate at the lowest point of his swing is approximately 18.70 meters per second.